Graham’s number

The World Champion largest number, listed in the latest
Guinness Book of Records,
is an upper bound, derived by R. L. Graham,
from a problem in a part of combinatorics called Ramsey theory.

Graham’s number cannot be expressed using the conventional
notation of powers, and powers of powers. If all the material in
the universe were turned into pen and ink it would not be enough
to write the number down. Consequently, this special notation,
devised by Donald Knuth,
is necessary.

3^3 means ‘3 cubed’, as it often does in computer printouts.

3^^3 means 3^(3^3), or 3^27, which is already quite large: 3^27
= 7,625,597,484,987, but is still easily written, especially as a
tower of 3 numbers: 333.

3^^^^3 = 3^^^(3^^^3), of course. Even the tower of exponents is
now unimaginably large in our usual notation, but Graham’s number
only starts here.

Consider the number 3^^^...^^^3 in which there are 3^^^^3
arrows. A largish number!

Next construct the number 3^^^...^^^3 where the number of arrows
is the previous 3^^^...^^^3 number.

An incredible, ungraspable number! Yet we are only two steps
away from the original ginormous 3^^^^3. Now continue this
process, making the number of arrows in 3^^^...^^^3 equal to the
number at the previous step, until you are 63 steps, yes,
sixty-three,
steps from 3^^^^3. That is Graham’s number.

There is a twist in the tail of this true fairy story. Remember
that Graham’s number is an upper bound, just like
Skewes’ number.
What is likely
to be the actual answer to Graham’s problem? Gardner quotes the
opinions of the experts in Ramsey theory, who suspect that the
answer is: 6.

A brief description of
the problem
for which Graham’s Number is an upper bound.
It also states that “More recently, Exoo (2003) has shown that must be
at least 11 and provides experimental evidence suggesting that it is actually even larger.”
But, presumably, nowhere near as large as G.

The wikipedia article
states that the lower bound was improved “to 13 by Jerome Barkley in 2008”.